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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 4, APRIL 2003 993 Geometrically Uniform Frames Yonina C. Eldar, Member, IEEE, and Helmut Bölcskei, Senior Member, IEEE Abstract—We introduce a new class of finite-dimensional frames with strong symmetry properties, called geometrically uniform (GU) frames, that are defined over a finite Abelian group of unitary matrices and are generated by a single generating vector. The notion of GU frames is then extended to compound GU (CGU) frames which are generated by a finite Abelian group of unitary matrices using multiple generating vectors. The dual frame vectors and canonical tight frame vectors associated with GU frames are shown to be GU and, therefore, also generated by a single generating vector, which can be computed very efficiently using a Fourier transform (FT) defined over the generating group of the frame. Similarly, the dual frame vectors and canonical tight frame vectors associated with CGU frames are shown to be CGU. The impact of removing single or multiple elements from a GU frame is considered. A systematic method for constructing optimal GU frames from a given set of frame vectors that are not GU is also developed. Finally, the Euclidean distance properties of GU frames are discussed and conditions are derived on the Abelian group of unitary matrices to yield GU frames with strictly positive distance spectrum irrespective of the generating vector. Index Terms—Compound geometrically uniform (CGU) frames, generalized Fourier transform (FT), geometrically uniform (GU) frames, least squares. I. INTRODUCTION F RAMES are generalizations of bases which lead to redun- dant signal expansions [1], [2]. A finite frame for a Hilbert space is a set of vectors that are not necessarily linearly in- dependent and span . Since the frame vectors can be linearly dependent, the conditions on frame vectors are usually not as stringent as the conditions on bases, allowing for increased flex- ibility in their design [3], [4]. Frames were first introduced by Duffin and Schaeffer [1] in the context of nonharmonic Fourier series, and play an impor- tant role in the theory of nonuniform sampling [1], [2], [5] and wavelet theory [3], [6]. Recently, frames have been used to ana- lyze and design oversampled filter banks [7]–[9] and error-cor- rection codes [10]. Frames have also been applied to the de- velopment of modern uniform and nonuniform sampling tech- niques [11], to various detection problems [12], [13], and to multiple description source coding [14]. Two important classes of highly structured frames are Gabor (Weyl–Heisenberg (WH)) frames [15], [16] and wavelet frames [3], [6], [17]. Both classes of frames are generated by a single Manuscript received August 12, 2001; revised December 3, 2002. Y. C. Eldar was with the Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139 USA. She is now with the Tech- nion–Israel Institute of Technology, Haifa 32000, Israel (e-mail: yonina@ee. technion.ac.il). H. Bölcskei is with the Swiss Federal Institute of Technology (ETH) Zurich, Communication Technology Laboratory, CH-8092 Zurich (e-mail: [email protected]). Communicated by G. Battail, Associate Editor At Large. Digital Object Identifier 10.1109/TIT.2003.809602 generating function. WH frames are obtained by translations and modulations of the generating function (referred to as the window function), and wavelet frames are obtained by shifts and dilations of the generating function (referred to as the mother wavelet). In Section III of this paper, we introduce a new class of frames which we refer to as geometrically uniform (GU) frames, that like WH and wavelet frames are generated from a single generating vector. These frames are defined by a finite Abelian group of unitary matrices, referred to as the generating group of the frame. We note that WH frames and wavelet frames are, in general, not GU since the underlying group of matrices is, in general, not Abelian. GU frames are based on the notion of GU vector sets first introduced by Slepian [18] and later extended by Forney [19], which are known to have strong symmetry proper- ties that may be desirable in various applications such as channel coding [19], [20], [21]. The notion of GU frames is then extended to frames that are generated by a finite Abelian group of unitary matrices using multiple generating vectors. Such frames are not necessarily GU, but consist of subsets of GU vector sets that are each gen- erated by . We refer to this class of frames as compound GU (CGU) frames, and develop their properties in Section VI. CGU frames are a generalization of filter-bank frames introduced for in [7]–[9]. An interesting class of frames results when the set of generating vectors is itself GU, generated by a finite Abelian group . (Note that this class of frames will in general not be GU.) As we show, these frames are a generalization of WH frames in which is the group of translations and is the group of modulations. Given a frame for , any signal in can be represented as a linear combination of the frame vectors. However, if the frame vectors are linearly dependent, then the coefficients in this ex- pansion are not unique. A popular choice of coefficients are the inner products of the signal with a set of analysis frame vectors called the dual frame vectors [17]. This choice of coefficients has the property that among all possible coefficients it has the minimal -norm [17], [22]. In Section IV, we show that the dual frame vectors associ- ated with a GU frame are also GU, and therefore generated by a single generating vector. Furthermore, we demonstrate that the generating vector can be computed very efficiently using a Fourier transform (FT) defined over the generating group of the frame. Similarly, in Section VI, we show that the dual-frame vectors associated with a CGU frame are also CGU. When the generating vectors of the CGU frame are GU and generated by a group that commutes up to a phase factor with the group , the dual frame is generated by a single generating vector, a result well known for WH frames. An important topic in frame theory is the behavior of a frame when elements of the frame are removed. In Section VII, we 0018-9448/03$17.00 © 2003 IEEE

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Page 1: Geometrically uniform frames - Information Theory, IEEE ... · Geometrically Uniform Frames Yonina C. Eldar, Member, IEEE, and Helmut Bölcskei, Senior Member, IEEE Abstract— We

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 4, APRIL 2003 993

Geometrically Uniform FramesYonina C. Eldar, Member, IEEE,and Helmut Bölcskei, Senior Member, IEEE

Abstract—We introduce a new class of finite-dimensionalframes with strong symmetry properties, called geometricallyuniform (GU) frames, that are defined over a finite Abelian groupof unitary matrices and are generated by a single generatingvector. The notion of GU frames is then extended tocompoundGU (CGU) frames which are generated by a finite Abelian groupof unitary matrices using multiple generating vectors. The dualframe vectors and canonical tight frame vectors associated withGU frames are shown to be GU and, therefore, also generated bya single generating vector, which can be computed very efficientlyusing a Fourier transform (FT) defined over the generating groupof the frame. Similarly, the dual frame vectors and canonical tightframe vectors associated with CGU frames are shown to be CGU.The impact of removing single or multiple elements from a GUframe is considered. A systematic method for constructing optimalGU frames from a given set of frame vectors that are not GU isalso developed. Finally, the Euclidean distance properties of GUframes are discussed and conditions are derived on the Abeliangroup of unitary matrices to yield GU frames with strictly positivedistance spectrum irrespective of the generating vector.

Index Terms—Compound geometrically uniform (CGU) frames,generalized Fourier transform (FT), geometrically uniform (GU)frames, least squares.

I. INTRODUCTION

FRAMES are generalizations of bases which lead to redun-dant signal expansions [1], [2]. A finite frame for a Hilbert

space is a set of vectors that are not necessarily linearly in-dependent and span. Since the frame vectors can be linearlydependent, the conditions on frame vectors are usually not asstringent as the conditions on bases, allowing for increased flex-ibility in their design [3], [4].

Frames were first introduced by Duffin and Schaeffer [1] inthe context of nonharmonic Fourier series, and play an impor-tant role in the theory of nonuniform sampling [1], [2], [5] andwavelet theory [3], [6]. Recently, frames have been used to ana-lyze and design oversampled filter banks [7]–[9] and error-cor-rection codes [10]. Frames have also been applied to the de-velopment of modern uniform and nonuniform sampling tech-niques [11], to various detection problems [12], [13], and tomultiple description source coding [14].

Two important classes of highly structured frames are Gabor(Weyl–Heisenberg (WH)) frames [15], [16] and wavelet frames[3], [6], [17]. Both classes of frames are generated by a single

Manuscript received August 12, 2001; revised December 3, 2002.Y. C. Eldar was with the Research Laboratory of Electronics, Massachusetts

Institute of Technology, Cambridge, MA 02139 USA. She is now with the Tech-nion–Israel Institute of Technology, Haifa 32000, Israel (e-mail: [email protected]).

H. Bölcskei is with the Swiss Federal Institute of Technology (ETH)Zurich, Communication Technology Laboratory, CH-8092 Zurich (e-mail:[email protected]).

Communicated by G. Battail, Associate Editor At Large.Digital Object Identifier 10.1109/TIT.2003.809602

generating function. WH frames are obtained by translationsand modulations of the generating function (referred to as thewindow function), and wavelet frames are obtained by shifts anddilations of the generating function (referred to as the motherwavelet). In Section III of this paper, we introduce a new class offrames which we refer to asgeometrically uniform (GU) frames,that like WH and wavelet frames are generated from a singlegenerating vector. These frames are defined by a finite Abeliangroup of unitary matrices, referred to as the generating groupof the frame. We note that WH frames and wavelet frames are,in general, not GU since the underlying group of matrices is, ingeneral, not Abelian. GU frames are based on the notion of GUvector sets first introduced by Slepian [18] and later extended byForney [19], which are known to have strong symmetry proper-ties that may be desirable in various applications such as channelcoding [19], [20], [21].

The notion of GU frames is then extended to frames that aregenerated by a finite Abelian groupof unitary matrices usingmultiple generating vectors. Such frames are not necessarilyGU, but consist of subsets of GU vector sets that are each gen-erated by . We refer to this class of frames ascompound GU(CGU) frames,and develop their properties in Section VI. CGUframes are a generalization of filter-bank frames introduced for

in [7]–[9]. An interesting class of frames results whenthe set of generating vectors is itself GU, generated by a finiteAbelian group . (Note that this class of frames will in generalnot be GU.) As we show, these frames are a generalization ofWH frames in which is the group of translations andis thegroup of modulations.

Given a frame for , any signal in can be represented as alinear combination of the frame vectors. However, if the framevectors are linearly dependent, then the coefficients in this ex-pansion are not unique. A popular choice of coefficients are theinner products of the signal with a set of analysis frame vectorscalled the dual frame vectors [17]. This choice of coefficientshas the property that among all possible coefficients it has theminimal -norm [17], [22].

In Section IV, we show that the dual frame vectors associ-ated with a GU frame are also GU, and therefore generated bya single generating vector. Furthermore, we demonstrate thatthe generating vector can be computed very efficiently using aFourier transform (FT) defined over the generating groupofthe frame. Similarly, in Section VI, we show that the dual-framevectors associated with a CGU frame are also CGU. When thegenerating vectors of the CGU frame are GU and generated bya group that commutes up to a phase factor with the group

, the dual frame is generated by asinglegenerating vector, aresult well known for WH frames.

An important topic in frame theory is the behavior of a framewhen elements of the frame are removed. In Section VII, we

0018-9448/03$17.00 © 2003 IEEE

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994 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 4, APRIL 2003

show that the frame bounds of the frame resulting from re-moving a single vector of a GU frame are the same regardless ofthe particular vector removed. In this sense, GU frames exhibitan interesting robustness property which is of particular impor-tance in applications such as multiple description source coding[23], [14]. We also consider the behavior of a GU frame whengroups of frame elements are removed.

In the special case of a tight frame, the dual frame vectorsare proportional to the original frame vectors so that the recon-struction formula is particularly simple. In many applications itis therefore desirable to construct a tight frame from an arbi-trary set of frame vectors. A popular tight-frame construction isthe so-called canonical tight frame [17], [7], [24]–[27], first pro-posed in the context of wavelets [28]. The canonical tight frameis relatively simple to construct, it is optimal in a least-squaressense [29], [27], [30], it can be determined directly from thegiven vectors, and plays an important role in wavelet theory[31]–[33]. Like the dual-frame vectors, we show that the canon-ical tight-frame vectors associated with a GU frame are GU, andthe canonical tight-frame vectors associated with a CGU frameare CGU. When the generating vectors of the CGU frame areGU and generated by a groupthat commutes up to a phasefactor with , the canonical tight-frame vectors can be obtainedby a single generating vector, generalizing a result well-knownin WH frame theory.

Since GU frames have nice symmetry properties, it may bedesirable to construct such a frame from a given set of framevectors. The problem of frame design has received relativelylittle attention in the literature. Systematic methods for con-structing optimal tight frames have been considered [29], [27],[30]. Methods for generating frames starting from a given frameare described in [4].

In Section VIII, we systematically construct optimal GUframes from a given set of vectors, that are closest in aleast-squares sense to the original frame vectors. The results inthis section rely on ideas developed in [34] in the context ofgeneral least-squares inner product shaping. In our develop-ment, we consider three different constraints on the GU framevectors. First, we treat the case in which the inner productsof the frame vectors are known. The optimizing frame isreferred to as the scaled-constrained least-squares GU frame(SC-LSGUF). Next, we consider the case where the innerproducts are known up to a scale factor. The optimizing framein this case is referred to as the constrained least-squares GUframe (C-LSGUF). Finally, we consider the case in which boththe inner products and the scaling are chosen to minimize theleast-squares error between the original frame and the resultingtight frame. The optimizing frame is the least-squares GUframe (LSGUF).

In Section IX, we consider distance properties of GU frames,which may be of interest when using GU frames for code design(group codes) [18], [19]. In particular, we introduce a class ofGU frames with strictly positive distance spectra for all choicesof generating vectors. Such GU frames are shown to be gener-ated by fixed-point-free groups [35].

Before proceeding to the detailed development, in Section II,we provide a brief introduction to frame expansions.

Fig. 1. Example of a frame. The vectors� , � , and � span IR and,therefore, form a frame forIR .

II. FRAMES

Frames, which are generalizations of bases, were introducedin the context of nonharmonic Fourier series by Duffin andSchaeffer [1] (see also [2]). Recently, the theory of frames hasbeen expanded [3], [6], [17], [4], in part due to the utility offrames in analyzing wavelet decompositions.

Let denote a set of complex vectors in an-dimensional Hilbert space . The vectors form a frame

for if there exist constants and such that1

(1)

for all [17]. In this paper, we restrict our attention to thecase where and are finite. The lower bound in (1) ensuresthat the vectors span ; thus, we must have . Since

, the right-hand inequality of (1) is always satisfied with, so that any finite set of vectors that spans

is a frame for . In particular, any basis for is a framefor . However, in contrast to basis vectors which are linearlyindependent, frame vectors with are linearly dependent.If the bounds in (1), then the frame is called atightframe. If, in addition, , then the frame is called anormalized tight frame. The redundancy of the frame is definedas , i.e., vectors in an -dimensional space.

A classical example of a frame is the frame depicted in Fig. 1.Since the vectors clearly span , they form a framefor . Note that the vectors are linearly dependent and there-fore do not constitute a basis for . The frame of Fig. 1 hasan interesting symmetry property: the frame vectors can be ob-tained by rotating any one of the vectorsby multiples of 120.As we will see in Section III, this frame is a cyclic frame whichis a special case of a GU frame.

The frame operatorcorresponding to the frame vectorsis defined as [17]

(2)

1We use the notationhx; yi to denote the scalar productx y, where vectorsxandy are represented as column vectors and� denotes the Hermitian transpose.

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ELDAR AND BÖLCSKEI: GEOMETRICALLY UNIFORM FRAMES 995

where is the matrix of columns , and denotes the Her-mitian transpose. Using the frame operator, (1) can be rewrittenas

(3)

for all .From (3), it follows that the tightest possible frame boundsand are given by and ,

where are the eigenvalues of the frameoperator . Throughout the paper, when referring to “the framebounds” we implicitly assume the tightest possible framebounds unless otherwise stated. Note that since the vectorsspan , is invertible so that .

If the vectors form a frame for , thenany can be expressed as a linear combination of thesevectors: . If , then the coefficients in thisexpansion are not unique. A possible choice iswhere are thedual frame vectors[17] of the frame vectors

, and are given by

(4)

The choice of coefficients has the property thatamong all possible coefficients it has the minimal-norm [17],[22].

There are other choices of dual frame vectorssuch that forany , . Specifically, with denotingthe matrix of columns , any other choice corresponds toofthe form [36]

(5)

where is an arbitrary matrix. Indeed, for any choiceof . However, the particular choice has some desir-able properties. Besides resulting in the minimal-norm coef-ficients, in many cases the choice yields frame vectorsthat share the same symmetries as the original frame vectors.Specifically, in Section IV we show that the dual frame vectorsassociated with a GU frame are also GU, and in Section VI weshow that the dual frame vectors associated with a CGU frameare also CGU. Finally, in the case of a tight frame, the dual framevectors lead to a particularly simple expansion. Specifically, inthis case so that , and the dual framevectors are . Since a tight frameexpansion of a signal is very simple, it is popular in many ap-plications [17].

Suppose we are given a set of vectorsthat form a frame for , with frame bounds . It maythen be desirable to construct a tight frame from these vectors.A popular tight-frame construction is the canonical tight frame[17], [7], [24], [25], [27], [29], first proposed in the context ofwavelets in [28]. Thecanonical tight-frame vectors

associated with the vectors are givenby

(6)

where is the symmetric positive–definite square root of. Note, that with an arbitrary unitary matrix

yields a tight frame as well. The canonical tight frame, however,has the property that it is the closest normalized tight frame tothe vectors in a least-squares sense [26], [29], [27].

From (4) and (6), we see that in order to compute the dual-frame vectors and the canonical tight-frame vectors associatedwith a frame , we need to compute the matrices and

and then apply them to each of the frame vectors.In the next section, we introduce a class of frames that havestrong symmetry properties calledGU frames. As we show inSection IV, the dual frame vectors and the canonical tight framevectors associated with a GU frame are generated by a singlegenerating function, and can therefore be computed very effi-ciently.

III. GU FRAMES

A. Definition

A set of vectors is GU [19], [18], [26]if every vector in the set has the form , where is anarbitrarygenerating vectorand the matricesare unitary and form an Abelian group2 . For concreteness,we assume that so that . The group will becalled thegenerating groupof .

Alternatively, a vector set is GU if given any two vectorsand in the set, there is an isometry (a norm-preserving lineartransformation) that transforms into while leaving theset invariant [19]. Thus, for every, . Intuitively, avector set is GU if it “looks the same” geometrically from anyof the points in the set.

The vector set of Fig. 1 is GU, since the set is symmetric withrespect to a rotation by 120. Further examples of GU vectorsets are considered in [19].

A set of vectors forms a GU frame forif the vectors are GU and span .

B. Properties of GU Frames

As we show in the following proposition, the frame boundsof a GU frame can be bounded by the norm of the generatingvector.

Proposition 1: Let be a GU framewith frame bounds and , where is an arbitrary generatingvector. Then, . If, in addition, the frame istight, then .

Proof: We can express the frame operator correspondingto the frame vectors as

(7)

Then

(8)

2That is,Q contains the identity matrixI ; if Q containsU , then it also con-tains its inverseU ; the productU U of any two elements ofQ is again inQ; andU U = U U for any two elements inQ [37].

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996 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 4, APRIL 2003

so that

(9)

Therefore,

(10)

and

(11)

Since , the inner product of two vectors in is

(12)

where is the function on defined by

(13)

For fixed , the set is just apermutation of since for all [37]. There-fore, the numbers are a permu-tation of the numbers . The same is truefor fixed . Consequently, every row and column of theGram matrix is a permutation of the numbers

.A matrix whose rows (columns) are a permutation of the

first row (column) will be called apermuted matrix.3 Thus, wehave shown that the Gram matrix of a GU vector set is a per-muted matrix. Furthermore, if the Gram matrixis a permuted matrix and in addition , then the vectors

are GU [34]. We, therefore, have the following proposi-tion.

Proposition 2: The Gram matrix corre-sponding to a GU vector set is a per-muted matrix. Conversely, if the Gram matrixis a permuted matrix, and for all , thenthe vectors are GU. If, in addition, the vectors span

, then they form a GU frame for .

As we will see in the sequel, the FT matrix plays an importantrole in defining GU frames. To define the FT it will be conve-nient to replace the multiplicative groupby an additive group

to which is isomorphic.4 Specifically, it is well known(see, e.g., [37]) that every finite Abelian group is isomor-phic to a direct product of a finite number of cyclic groups:

, where is the cyclic additivegroup of integers modulo , and . Thus, every ele-

3An example of a permuted matrix is

a a a a

a a a a

a a a a

a a a a

: (14)

4Two groupsQ andQ are isomorphic,denoted byQ �= Q , if there is a

bijection (one-to-one and onto map)': Q ! Q which satisfies'(xy) ='(x)'(y) for all x, y 2 Q [37].

ment can be associated with an element of theform , where ; this correspon-dence is denoted by .

Each vector is then denoted as , where. The zero element corresponds to

the identity matrix , and an additive inversecorresponds to a multiplicative inverse . TheGram matrix is then the matrix

(15)

with row and column indexes , where is now thefunction on defined by

(16)

The FT of a complex-valued function: defined onis the complex-valued function:

defined by

(17)

where the Fourier kernel is

(18)

Here, and are the th components of and , respectively,and the product is taken as an ordinary integer modulo.

The FT matrix over is defined as the matrix

The FT of a column vector is then thecolumn vector given by . Sinceis unitary, we obtain the inverse FT formula

(19)

As we show in the following theorem, the FT matrix can beused to define GU frames.

Theorem 1: A set of vectors in an -di-mensional Hilbert space is GU if and only if the Gram matrix

is diagonalized by an FT matrix over a finiteproduct of cyclic groups . The vectors form a GU framefor if in addition has rank .

Proof: The vectors form a frame for if and onlyif they span , which implies that the rank of , must be equalto .

For a GU vector set with generating group , the FTover diagonalizes the Gram matrix [26]. Thus, to completethe proof of the theorem, we need to prove that ifis diago-nalized by an FT matrix over the group , then the vector set

is GU.Let be the matrix of columns , so that . Sincediagonalizes , has an eigendecomposition of the form

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ELDAR AND BÖLCSKEI: GEOMETRICALLY UNIFORM FRAMES 997

for a diagonal matrix with diagonal elements, where the first diagonal elements may be nonzero and

the remaining diagonal elements are all zero. Then,has a sin-gular value decomposition (SVD) [38] of the form ,where is an arbitrary unitary matrix and is an diag-onal matrix with diagonal elements .

Let denote the columns of . From the defi-nition of , the components of are all equal to , and

where is a diagonal unitary matrix withdiagonal elements , where is given by(18). Then

(20)

where , and where we used the fact that diagonalmatrices commute. If we now define , thenwe have that , where the matricesare unitary.

We now show that the group is anAbelian group. First, we have that

so that . Next

so that since . Finally,

since diagonal matrices commute, and sincefor some .

We, therefore, conclude that where the ma-trices are unitary and form an Abelian group, so that thevectors are GU.

As a consequence of Theorem 1 we have the following corol-lary.

Corollary 1: A set of vectors is GUif and only if the matrix of columns has an SVD of theform , where is an arbitrary unitary matrix, isan arbitrary diagonal matrix with diagonal elements

, and is an FT matrix over a direct product of cyclicgroups. In addition, the vectors form a GU frame for ifthey span or, equivalently, if for .

IV. DUAL AND CANONICAL TIGHT FRAMES ASSOCIATED

WITH GU FRAMES

In Section IV-A, we show that the dual-frame vectors and thecanonical tight-frame vectors associated with a GU frame arealso GU. This property can then be used to compute the dual andcanonical tight frames very efficiently. Further properties of thecanonical tight-frame vectors are discussed in Section IV-B.

A. Constructing the Dual and Canonical Tight Frames

Let be a GU frame generatedby a finite (not necessarily Abelian) group of unitary ma-trices, where is an arbitrary generating vector. Then the frame

operator defined by (2) commutes with each of the unitarymatrices in the generating group . Indeed, expressing theframe operator as

(21)

we have that for all

(22)

since is just a permutation of .If commutes with , then and also commute

with for all . Thus,

(23)

where , which shows that the dual frame vectorsare GU with generating group equal to.

Similarly

(24)

where , which shows that the canonical tight framevectors are also GU with generating group.

Therefore, to compute the dual frame vectors or the canonicaltight frame vectors all we need is to compute the generatingvectors and , respectively. The remaining frame vectors arethen obtained by applying the group to the correspondinggenerating vectors.

We now show that when the group is Abelian, the gen-erating vectors can be computed very efficiently using the FT.From Corollary 1, we have that has an SVD of the form

(25)

Here, is a diagonal matrix with diagonal elements

where is the FT of , is thematrix of columns , where

ifotherwise

(26)

with

(27)

denoting the th element of the FT of regarded as a row vectorof column vectors, , and

has rows .

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998 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 4, APRIL 2003

It then follows that

(28)

where if . Similarly

(29)

We summarize our results in the following theorem.

Theorem 2 (GU Frames):Let bea GU frame generated by a finite Abelian groupof unitarymatrices, where is an arbitrary generating vector, and letbethe matrix of columns . Let be an additive Abelian groupisomorphic to , let be the elements of underthis isomorphism, and let be the FT matrix over . Then

1) the dual frame vectors are GU withgenerating group and generating vector

where

a) are the singularvalues of ;

b) is the FT of the inner-product se-quence ;

c) is the set of indexes for which ;d) for ;e) is the FT of ;

2) the canonical tight frame vectors areGU with generating group and generating vector

;

3) the frame bounds of the frame are givenby and .

An important special case of Theorem 2 is the case in whichthe generating group is cyclic so thatwhere is a unitary matrix with . A cyclic group gen-erates a cyclic vector set ,where is arbitrary. For example, the frame in Fig. 1 is cyclicwith denoting a rotation by 120. If is cyclic, then is acirculant matrix,5 and is the cyclic group . The FT kernelis then for , and the FT matrix

reduces to the discrete FT (DFT) matrix. The singularvalues of are then times the square roots of the DFTvalues of the inner products .

B. Properties of the Canonical Tight Frame

The canonical tight-frame vectors corresponding to theframe-vectors have the property that they are the closest nor-malized tight-frame vectors to the vectors, in a least-squares

5A circulant matrix is a matrix where every row (or column) is obtained bya right circular shift (by one position) of the previous row (or column). An ex-ample is

a a a

a a a

a a a

:

sense [29], [27], [30]. Thus, the vectors are the normalizedtight-frame vectors that minimize the least-squares error

(30)

We now show that when the original frame vectorsare GUwith generating group , the canonical tight-frame vectors havethe additional property that among all normalized tight-framevectors they maximize

(31)

Maximizing may be of interest in various applications.For example, in a matched-filter detection problem consideredin [12], represents the total output signal-to-noise ratio. Asanother example, in a multiuser detection problem considered in[13], maximizing has the effect of minimizing the multiple-access interference at the input of the proposed detector.

To obtain a more convenient expression for , let anddenote the matrices of columnsand , respectively. Since thevectors form a normalized tight frame for , satisfies

(32)

From Corollary 1, has an SVD of the form , whereis unitary, is the FT matrix over the additive group to

which is isomorphic, and is an diagonal matrix withdiagonal elements . From (32), it follows that can bewritten as where is an arbitrary unitary matrixand is an diagonal matrix with diagonal elements allequal to .

Let and denote the columns of and , respectively.Then we can express as

(33)

where is an diagonal matrix with the first diagonalelements equal to , and the remaining diagonal elements areall equal to .

Our problem then reduces to finding a set of orthonormalvectors that maximize , where the vectorsare also orthonormal. Using the Cauchy–Schwarz inequality, wehave that

(34)with equality if and only if for some . Inparticular, we have equality for . Since the componentsof the vectors all have equal magnitude ,

for all

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ELDAR AND BÖLCSKEI: GEOMETRICALLY UNIFORM FRAMES 999

Fig. 2. Example of a GU frame.

and (34) reduces to

(35)with equality if .

The normalized tight-frame vectors that maximize arethen the columns of where ,and are equal to the canonical tight frame vectors.

V. EXAMPLE OF A GU FRAME

We now consider an example demonstrating the ideas of theprevious section.

Consider the frame vectors ,, , ,

depicted in Fig. 2.The corresponding Gram matrix is given by

(36)

which is a permuted matrix with . From Proposition 2it follows that the vectors are GU. Since the vectors alsospan , these vectors form a GU frame for .

The vectors can be expressed as ,where and the matrices are unitary,form an Abelian group , and are given by

(37)

The multiplication table of the group is

(38)

If we define the correspondence

(39)

then this table becomes the addition table of

(40)

Only the way in which the elements are labeled distinguishesthe table of (40) from the table of (38); thus,is isomorphic to

. Over , the FT matrix is the Hadamard matrix

(41)

From Theorem 2, the dual-frame vectors and the canonicaltight-frame vectors are also GU with generatorsand , re-spectively, whose expressions are given in the theorem. Thus,to compute the dual and canonical tight-frame vectors we com-pute these generators and then apply the group.

We first determine the FT of the first row of denoted by

(42)

Using Theorem 2, it follows from (42) that the frame bounds aregiven by and . Next, we compute the vectorswhich are the columns of

(43)

Using the expressions of the theorem, we then have thatand . By applying the group to

these generators we obtain that the dual-frame vectors are thecolumns of

(44)

and the canonical tight-frame vectors are the columns of

(45)

Comparing (44) and (45) with the original frame vectors,it is evident that the dual and canonical tight-frame vectors havethe same symmetries as the original frame vectors, as illustratedin Fig. 3.

VI. COMPOUND GU FRAMES

In Section IV, we showed that the dual and canonical tight-frame vectors associated with a GU frame are themselves GUand can, therefore, be computed using a single generator. In thissection, we consider a class of frames which consist of subsetsthat are GU, and are, therefore, referred to ascompound GU(CGU) frames. As we show, the dual and canonical tight-framevectors associated with a CGU frame share the same symme-tries as the original frame and can be computed using asetofgenerators.

A set of frame vectors is CGUif for some generating vectors ,

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Fig. 3. Symmetry property of the frame vectors� , the dual-frame vectors� , and the canonical tight-frame vectors� . � are the columns of� givenby (44), and� are the columns ofM given by (45). The frame vectors,dual-frame vectors, and the canonical tight-frame vectors all have the samesymmetry properties.

and the matrices are unitary and form anAbelian group .

A CGU frame is in general not GU. However, for every, thevectors are a GU vector set with generatinggroup .

A special case of CGU frames are filter-bank frames studiedfor in [7]–[9], in which is the group of translations byinteger multiples of the subsampling factor, and the generatingvectors are the filter-bank synthesis filters.

As we show in the following proposition, the frame boundsof a CGU frame can be bounded by the sum of the norms of thegenerating vectors.

Proposition 3: Letbe a compound GU frame with frame boundsand , where

is an arbitrary set of generating vectors. Then

If, in addition, the frame is tight, then

Proof: We can express the frame operator correspondingto the frame vectors as

(46)

Then

(47)

so that

(48)

Fig. 4. A compound GU frame. The setsS = f� ; � g and S =

f� ; � g are both GU with the same generating group; both sets areinvariant under a reflection about the dashed line. However, the combined setS = f� ; � ; � ; � g is no longer GU.

Therefore,

(49)

and

(50)

A. Example of a CGU Frame

An example of a CGU frame is illustrated in Fig. 4. In thisexample, the frame vectors are where

, with

(51)

and the generating vectors are

(52)

The matrix represents a reflection about the dashed line inFig. 4. Thus, the vector is obtained by reflecting the gener-ator about this line, and similarly, the vector is obtainedby reflecting the generator about this line.

As can be seen from the figure, the frame is not GU. In par-ticular, there is no isometry that transforms into whileleaving the set invariant. However, the setsand are both GU with generating group.

B. Dual and Canonical Tight Frames Associated With CGUFrames

We now show that the dual and canonical tight frames asso-ciated with a CGU frame are also CGU.

Expressing the frame operator as

(53)

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ELDAR AND BÖLCSKEI: GEOMETRICALLY UNIFORM FRAMES 1001

for all we have that

(54)

since is just a permutation of . Thus,commutes with , so that and also commute with

for all . Then, the dual-frame vectors of the vectorsare given by

(55)

where , which shows that the dual-frame vectorsare CGU with generating group equal to.

Similarly

(56)

where , which shows that the canonical tight-frame vectors are also CGU with generatinggroup .

Therefore, to compute the dual-frame vectors or the canonicaltight-frame vectors all we need is to compute the generatingvectors and , respectively.The remaining frame vectors are then obtained by applying thegroup to the corresponding set of generating vectors.

For the CGU set of Fig. 4 we have that

(57)

Therefore, and. Since in this example the generating vectors of

the dual frame and the canonical frame are proportional to thegenerating vectors of the original frame, the dual-frame vectorsand the canonical frame vectors are proportional to the originalframe vectors.

C. CGU Frames With GU Generators

A special class of CGU frames isCGU frames with GU gen-erators in which the generating vectors arethemselves GU. Specifically, for some generator

, where the matrices are unitary, and forman Abelian group .

Now suppose that and commute up to a phase factorfor all and so that where is anarbitrary phase function that may depend on the indexesand .In this case, we say that and commute up to a phase factor.Then for all ,

(58)

The dual frame vectors of the vectors are then given by

(59)

where . Similarly

(60)

where . Thus, even though the frame is not in gen-eral GU, the dual and canonical tight-frame vectors can be com-puted using a single generating vector.

Alternatively, we can express as where. Similarly, where . It then follows

that the generators and are both GU with generating group.

We conclude that for a CGU frame with commuting GU gen-erators and generating group, the dual frame and the canon-ical frame are also CGU with commuting GU generators andgenerating group .

As we now show, in the special case in which sothat for all , the resulting frame is GU.To this end, we need to show that the unitary matrices

form an Abelian group. First,

Since and , . Next,

since and . Also, since and. Finally,

A special case of CGU frames with GU generators for whichand commute up to a phase factor are the WH frames [17],

[3], [16]. If the WH frame is critically sampled, thenand the WH frame reduces to a GU frame. In the more generaloversampled case, .

To summarize, we have the following theorem.

Theorem 3 (CGU Frames):Let

be a CGU frame with generating vectorsand generating group, and let be the frame operator corre-sponding to the frame vectors . Then

1) the dual frame vectorsare CGU with generating group and generating vectors

;

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1002 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 4, APRIL 2003

2) the canonical tight-frame vectorsare CGU with generating group and generating

vectors .

If in addition, the generating vectorsare GU with for all , then

1) where so that the dual-framevectors are CGU with GU generators;

2) where so that the canonicalframe vectors are CGU with GU generators;

3) if in addition for all , then the vectorsform a GU frame.

VII. PRUNING GU FRAMES

In applications, it is often desirable to know how a frame be-haves when one or more frame elements are removed. In par-ticular, it is important to know or to be able to estimate theframe bounds of the reduced frame. In general, if no structuralconstraints are imposed on a frame, this behavior will dependcritically on the particular frame elements removed. For ex-ample, removal of a particular frame element may destroy theframe property so that the remaining vectors do not constitutea frame anymore, whereas if a different element is removed theremaining vectors may still constitute a frame.

One of the prime applications of frames is signal analysis andsynthesis, where a signal is expanded by computing the innerproducts of the signal with the frame elements. The resultingcoefficients are subsequently stored, transmitted, quantized, ormanipulated in some way. In particular, a coefficient may belost (e.g., due to a transmission error) which results in a recon-structed signal that is equivalent to an expansion using a prunedframe obtained by removing the corresponding frame vector.

Recently, there has been increased interest in using frames formultiple-description source coding where a signal is expandedinto a redundant set of functions and the resulting coefficientsare transmitted over a lossy packet network, where one or moreof the coefficients can be lost because a packet is dropped [23],[14]. The goal of multiple description source coding is to ensurea gradually behaving reconstruction quality as a function of thenumber of dropped packets. When using frames in this context,the reconstruction quality is often governed by the frame boundratio of the pruned frame. If the packets are dropped with equalprobability, then it is desirable that the frame bound ratio shoulddeteriorate uniformly irrespectively of the particular frame ele-ment that is removed. In the following, we show that GU frameshave this property. We, furthermore, demonstrate that if the orig-inal frame is a tight GU frame, then the frame bound ratio of thepruned frame obtained by removing one frame element can becomputed exactly. We also consider the case where sets of frameelements are removed.

Theorem 4 (Pruned GU Frames):Let

be a GU frame generated by a finite Abelian groupof unitarymatrices, where is an arbitrary generating vector. Letbe the

matrix of columns , and let be the correspondingframe operator. Let

be the pruned set obtained by removing the element. Then theeigenvalues of the frame operator corresponding to the prunedset do not depend on the particular elementremoved.

Proof: The frame operator corresponding to the prunedframe is given by

(61)

Since is unitary, the eigenvalues of are equal to theeigenvalues of . But

(62)

Since is independent of , the eigenvalues ofdo not depend on.

In general, it is difficult to provide estimates on the framebounds of the pruned frame. However, in the special case wherethe original GU frame is a tight frame, these bounds can bedetermined exactly.

Corollary 2 (Pruned Tight GU Frames):Let

be a GU tight frame generated by a finite Abelian groupofunitary matrices, where is a unit norm generating vector. Let

be the matrix of columns , and let be the corre-sponding frame operator. Let

be the pruned set obtained by removing the element. Then,the eigenvalues of the frame operator corresponding to thepruned set are given by and ,independent of .

A similar result was also shown by Goyalet al. [14, The-orem 4.1].

Proof: Since is a tight frame with , from Propo-sition 1, the frame bound and . Then

(63)

and

(64)

Since , has one eigenvalue equal to, and theremaining eigenvalues equal to. The eigenvalues of aretherefore given by and .

An immediate consequence of Corollary 2 is that the framebound ratio of the pruned frame is given by

, which is close to for large redundancy .We next consider the case where multiple frame elements are

removed.

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ELDAR AND BÖLCSKEI: GEOMETRICALLY UNIFORM FRAMES 1003

Corollary 3: Let be a GU framegenerated by a finite Abelian group of unitary matrices, let

be the matrix of columns , and let be the cor-responding frame operator. Let be a set of indexes, and let

denote the set of indexessuch that for fixedand . Let be a

pruned set obtained by removing the elementswith .Then, the eigenvalues of the frame operator corresponding to thepruned set are independent of.

Proof: The frame operator corresponding to the prunedframe is given by

(65)

Then

is independent of , and, consequently, the eigenvalues ofdo not depend on.

To conclude this section, GU frames have strong symmetryproperties in the sense that removing any one of the elementsleads to a vector set with bounds independent of the particularelement removed. Moreover, if the original frame is tight, thenwe can compute the bounds of the pruned frame exactly.

VIII. C ONSTRUCTINGGU FRAMES

Suppose we are given a set of vectors thatform a frame for an -dimensional space . We would like toconstruct a GU frame from the vectors .

From Theorem 1, it follows that the vectors form a GUframe if and only if the Gram matrix has rank , and is di-agonalized by an FT matrix over a finite product of cyclicgroups. There are many ways to construct a frame from a givenset of frame vectors that satisfy these properties. For ex-ample, let be the matrix of columns , and let have anSVD , where is a diagonal matrix with diagonalelements . Then, the columns ofform a GU frame, where is any FT matrix over a productof cyclic groups, and is an arbitrary diagonal matrix withdiagonal elements . The frame bounds of the resultingGU frame are given by and ,so that we can choose the diagonal matrixto control thesebounds. In particular, choosing we have that the columnsof form a GU frame. This choice has theproperty that the frame bounds of the GU frame are equal to theframe bounds of the original frame.

We now consider the problem of constructing anoptimalGUframe. Specifically, let denote a frame for ,and suppose we wish to construct a GU frame from thevectors . A reasonable approach is to find a set of vectors

that span , and are “closest” to the vectors in the least-

squares sense. Thus, we seek vectorsthat minimize the least-squares error , defined by

(66)

where denotes theth error vector

(67)

subject to the constraint that the vectorsform a GU frame.If the vectors are GU, then their Gram matrix is

a permuted matrix with rank , diagonalized by an FT matrix. Thus, the inner products must satisfy

(68)

where is a permutation of the numbers, is a scaling factor, and the numbersare chosen such that the matrix, whose th

row is equal to , is Hermitian, nonnegativedefinite, and diagonalized by an FT matrix.

In our development of the optimal GU frame vectors we as-sume that the permutations in (68) are specified. Since thesepermutations determine the additive groupover which the FTmatrix is defined, we assume that is specified. We thenconsider three different constraints on the vectors. First, weconsider the case in which both the numbersand the scaling in (68) are known. The GU frame minimizingthe least-squares error of (66) and (67) subject to this con-straint is derived in Section VIII-A, and is referred to as theSC-LSGUF. Next, we consider the case in which the numbers

in (68) are known, and the scalingis chosento minimize . The resulting GU frame is referred to as theC-LSGUF, and is derived in Section VIII-B. Finally, in Sec-tion VIII-C, we consider the more general case in which boththe numbers and the scaling in (68) arechosen to minimize . The resulting GU frame is referred to asthe least-squares GU frame (LSGUF).

A. Scaled-Constrained Least-Squares GU Frame (SC-LSGUF)

We first consider the case in which the numbersand the scaling in (68) are known. Thus, we seek the set

of vectors that minimize the least-squares errorof (66)and (67) subject to the constraint

(69)

where is the matrix of columns , is a known scalingfactor, and is the matrix whoseth row is equal to

, where the numbers are given such thatis diagonalized by and is a diagonal matrix with diagonal

elements whereis the FT of the sequence . From (69), theframe bounds of the vectors are given byand .

The optimal SC-LSGUF vectors follow from [34], and arethe columns of where

(70)

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1004 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 4, APRIL 2003

Here, and are the right-hand unitary matrix and left-handunitary matrix, respectively, in the SVD of , is the ma-trix of columns , and is an diagonal matrix withdiagonal elements for values of for which .

If is invertible, then we may expressas

(71)

B. Constrained Least-Squares GU Frame (C-LSGUF)

We now consider the case in which the numbersare known, but the scaling is not specified. Thus, we

seek a set of vectors that minimize the least-squares errorsubject to

(72)

where .The least-squares errorof (66) and (67) may be expressed

as

(73)

Let . Then minimizing is equivalent to mini-mizing

(74)

subject to

(75)

To determine the optimal matrix we have to minimizewith respect to and . Fixing and minimizing with respectto , the optimal value of is given by the SC-LSGUF of Sec-tion VIII-A with scaling , so that

(76)

If is invertible, then

(77)

Substituting back into (74), and minimizing with respect to, the optimal value of is

(78)

which in the case that is invertible reduces to

(79)

The C-LSGUF vectors are then the columns ofgiven by

(80)

where is given by (78). If is invertible, then

(81)

with given by (79).We summarize our results regarding constrained optimal GU

frames in the following theorem.

Theorem 5 (C-LSGUF):Let be a set of vectors inan -dimensional Hilbert space that span , and let bethe matrix of columns . Let denote the optimal GUframe vectors that minimize the least-squares error defined by(66) and (67), subject to the constraint (68), and letbe thematrix of columns . Let be the matrix with th row equalto , so that is the Gram matrix of innerproducts , and let be the FT matrix that diagonal-izes . Let be the diagonal matrix with diagonal elements

where is theFT of the sequence , let be an di-agonal matrix with diagonal elements for values of forwhich , and let and be the right-hand unitary matrixand left-hand unitary matrix, respectively, in the SVD of .Then

1) if is given, then . If in additionis invertible, then

2) if is chosen to minimize , then ,where is given by (78). If, in addition, is invert-ible, then

where is given by (79).

C. Least-Squares GU Frame (LSGUF)

We now consider the least-squares problem in which both thescaling factor and the numbers in (68) arechosen to minimize . Thus, we seek a set of vectors thatminimize the least-squares errorof (66) and (67) subject to

(82)

where is a known permutation of the un-known numbers , chosen such that the matrix

whose th row is equal to is Hermitian,nonnegative definite, and diagonalized by an FT matrix.

This problem has been considered in the context of generalleast-squares inner product shaping [34], in which it was shownthat the solution involves solving a problem of the form

(83)

subject to

(84)

where the vectors are known and are a function of the givenvectors .

As we now show, this problem is equivalent to a detectionproblem in quantum mechanics, for which there is no known an-alytical solution in the general case. However, there exist veryefficient computational methods for obtaining a numerical so-lution [39], [40].

1) Connection With Quantum Detection:In a quantum de-tection problem, a system is prepared in one ofknown (pure)states that are described by vectorsin a Hilbert space , andthe problem is to detect the state prepared by performing a mea-surement on the system. The measurement is described in termsof a set of orthogonal measurement vectors. Given a set of

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ELDAR AND BÖLCSKEI: GEOMETRICALLY UNIFORM FRAMES 1005

measurement vectors, and assuming equal prior probabilitieson the different states, the probability of detection is given by[41]

(85)

Comparing (85) with (83), we see that finding a set of or-thonormal measurement vectors to maximize the probabilityof detection is equivalent to the maximization problem of (83)and (84).

Necessary and sufficient conditions for an optimum measure-ment maximizing (85) have been derived [42], [43], [41], [39].However, except in some particular cases [41], [44], [45], [26],[46], obtaining a closed-form analytical expression for the op-timal measurement directly from these conditions is a difficultand unsolved problem. Efficient iterative algorithms for max-imizing (85) that are guaranteed to converge to the global op-timum are given in [39], [40].

We conclude that in general there is no known analytical ex-pression for the LSGUF. However, in practice the LSGUF canbe approximated to any desired accuracy very efficiently usingthe iterative algorithms of [39], [40].

IX. DISTANCE PROPERTIES OFGU FRAMES

So far, we have mainly been concerned with structural prop-erties of GU frames. In this section, we study the Euclidean dis-tance properties of these frames.

Suppose we are given a GU framegenerated by the group with . We would like tocharacterize the distance profile for all

.Since the vectors are GU, is just a

permutation of . Furthermore

(86)

where are the elements of the first row of theGram matrix corresponding to the frame .

In applications, it may be desirable to construct a GU framesuch that for . Since ,a sufficient condition is that

(87)

which is satisfied if and only if none of the matrices hasan eigenvalue equal to. Note that if (87) is satisfied, then

regardless of the generating vector .Groups with unitary representations satisfying (87) are knownasfixed-point free groups, and have been studied extensively inthe literature (see, e.g., [47]). Thus, ifis a representation of afixed-point free group, then we have that .

Fixed-point free groups have recently been studied in the con-text of unitary space–time codes [35]. In particular, it was shownin [35] that an Abelian group of matrices satisfies (87) ifand only if it is cyclic, i.e., with , and where

can be parameterized as

(88)

where is relatively prime to for all . An optimization overthe can be performed to obtain distance profiles with certainprescribed properties (there are positive integers less than

that are relatively prime to , where denotes the Eulertotient function of ).

X. CONCLUSION

In this paper, we introduced the concept of GU and CGUframes and discussed some of their key properties. A funda-mental characteristic of these frames is that they possess strongsymmetry properties that may be desirable in a variety of appli-cations. In particular, similar to WH frames and wavelet frames,GU frames are generated by a single generating vector. Further-more, the canonical and dual-frame vectors associated with aGU frame are themselves GU and are, therefore, also generatedby a single generating vector which can be computed very effi-ciently using an FT matrix defined over an appropriate group.

We also showed that GU frame vectors exhibit interestingsymmetry properties when one or more frame elements are re-moved. This property of GU frames may be of importance inmultiple description source coding where it is often desirablethat the quality of the reconstruction should not depend on theparticular elements lost (removed).

Although in this paper we have focused on the case in whichthe underlying group is a finite Abelian group, many of the re-sults can be extended to the more general case of infinite-di-mensional and non-Abelian groups. An interesting direction forfurther research is to characterize these more general cases ofGU frames using possibly continuous-time FTs defined overnon-Abelian groups (see, e.g., [48]).

ACKNOWLEDGMENT

The authors wish to thank Prof. G. D. Forney for many fruitfuldiscussions.

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